7 ms·

The disease-induced herd immunity level for Covid-19 is lower than the classical

34 points by 2a0c40 6y ago

rossdavidh 6y ago
Interesting that it seems to suggest there is an optimal amount of distancing/lockdown/etc. to lower the peak, but avoid a big "rebound". I don't see a lot of discussion about how to find that optimum amount. It has occurred to me that the planet is essentially conducting a gigantic experiment in virus evolution, by creating a novel viral selection environment never seen before. For example, how does it impact a virus if the strains most likely to send the host to the hospital, actually spread more than the strains most likely to be asymptomatic, because even asymptomatic hosts are in isolation? Unless there is a vaccine or cure coming soon (there isn't), then the end result must be herd immunity, no matter what the policy. It would seem that we should be considering how to select for the healthiest segment of the population to provide that herd immunity, rather than trying to isolate as much as possible in the vain hope that it will die out prior to that point. If that was every possible, it has long since spread way too far to expect that to be possible. There might be cases (e.g. MERS) where you can stomp out a virus before it spreads enough to cause herd immunity, but we have long since passed that point. I don't think most people (including policy makers) are thinking about the endgame properly (as this paper does).
- jakeogh 6y ago
  Senegal is worth looking at (end of vid): https://youtu.be/iBma_0oAiMI https://youtu.be/iBma_0oAiMI
- macintux 6y ago
  As far as I can tell, there is one big unknown that makes planning particularly difficult: What are the long-term health effects from being infected? If it turns out that some significant percentage of the population is going to have a negative impact over the course of their lives from this, should we continue to hunker down until a vaccine or better treatments are found?
  - irq11 6y ago
    This is not an unknown. We have very good data on outcomes. The media has been diving ever-further into niche stories of bad outcomes in order to drive clicks, but the numbers of cases driving these narratives are vanishingly small. For the vast majority (>98%) of those who are under 60 and otherwise healthy, this disease is on par with the flu. Even amongst those who are more vulnerable, the rate of long-term sequelae is low. It’s the people who end up in the ICU who are experiencing the most long-term effects, and those people are a single-digit percentage of a single-digit percentage of those infected.
    - macintux 6y ago
      This disease has been here for 6 months. The degree of certainty in your assessment seems unwarranted. I want to believe it, but optimism and certainty given the novelty of this virus seem strange bedfellows.
      - xamuel 6y ago
        As rossdavidh correctly pointed out, if we don't get a vaccine soon (which we probably won't), then herd immunity is virtually the only way out anyway. Even if it turns out there are long term health problems, that doesn't really change that calculus. Very few people can completely isolate themselves indefinitely (remember, the virus doesn't just magically not spread in supermarkets), and if everyone did that, most of us would starve and the few who didn't starve would inherit a post-civilization ruin.
        mlyle 6y ago
        It does change the economic break even point a little bit. If intervention A reduces infection count 10% in the time to a vaccine, whether it's worth it depends upon whether it prevents 20,000 deaths, or 20,000 deaths and 40,000 long term illnesses. The threshold to herd immunity depends upon contact rate. So interventions well short of "lockdown" may not completely prevent the spread of the virus but still reach equilibrium at a lower number of people infected.
- Isinlor 6y ago
  Currently we need to learn more about this virus to make informed decisions - there is more we still don't know, than we do know for sure. Trying to slowly contain the virus like in South Korea, Taiwan, New Zealand is a strategy worth a shot for most of the western world, unless we learn something dramatic new about it. There is no need for draconian lock downs after exponential spread is stopped, but we do need good track and trace programs, masks etc. There are very high risks associated with Swedish strategy of either killing a lot of people due to bad separation of vulnerable people, destroying health of the big part of the population (see below), or even just failing to reach herd immunity because social distancing, closure of schools, and ban on mass gathering will be good enough to slow down the spread a lot, but bad enough to kill disproportionate amount of people. For example Sweden has many times more deaths than other Nordic Countries, but similar economical outlooks, and the even the idea of reaching herd immunity is uncertain. It may be that other Nordic countries will have soon less or about the same amount of restrictions as Sweden, but many times less deaths. And regarding the risks, one of the scientists who discovered Ebola [0]: "Many people think COVID-19 kills 1% of patients, and the rest get away with some flulike symptoms. But the story gets more complicated. Many people will be left with chronic kidney and heart problems. Even their neural system is disrupted. There will be hundreds of thousands of people worldwide, possibly more, who will need treatments such as renal dialysis for the rest of their lives. The more we learn about the coronavirus, the more questions arise. We are learning while we are sailing. That’s why I get so annoyed by the many commentators on the sidelines who, without much insight, criticize the scientists and policymakers trying hard to get the epidemic under control. That’s very unfair." Also based on Japanese study on around 100 COVID-19 cases from Grand Princess passengers [1] and some 6 divers all with mild cases from Germany [2] around 50 to 80% of people seem to have damage to the lungs visible on CT scans after mild symptomatic or even asymptomatic infection. Two of the divers had significant oxygen deficiency when doing physical exercise after 5-6 weeks of recovery. The guess of the doctor who was inspecting them is that this is permanent damage that may take years to recover if ever. [0] https://www.sciencemag.org/news/2020/05/finally-virus-got-me-scientist-who-fought-ebola-and-hiv-reflects-facing-death-covid-19 https://www.sciencemag.org/news/2020/05/finally-virus-got-me... [1] https://pubs.rsna.org/doi/10.1148/ryct.2020200110 https://pubs.rsna.org/doi/10.1148/ryct.2020200110 [2] https://translate.google.com/translate?sl=auto&tl=en&u=https%3A%2F%2Fwww.wetnotes.eu%2Ftauchen-nach-covid-19-erkrankung%2F https://translate.google.com/translate?sl=auto&tl=en&u=https...
- deleted 6y ago
  [deleted]
medymed 6y ago
It seems like their simulated model makes certain members of a population less ‘active’ and so at less parametric risk of infection/transmission than in a model where everyone is active. This is useful, and/but doesn’t appear to be Covid-19 specific.
- SpicyLemonZest 6y ago
  It isn't, and the same phenomenon has been observed in other cases. The 2009 swine flu for example had R0 estimated between 1.4-1.6, yielding a theoretical herd immunity threshold of 29%-33%, but estimates of how many people were actually infected range from 11%-21%.
- jaynetics 6y ago
  I thought their point was that highly active people will have been infected in greater numbers after a lockdown, numbers that are even more divergent than without a lockdown, and thus less other people will need to get infected.
lostlogin 6y ago
> The classical herd immunity level hC is defined as hC=1−1/R0 How do you use this in places where transmission is less than 1? New Zealand now has a transmission rate considerably under 1, but this seems to break the calculation. Does this mean that a reproduction number under 1 doesn’t need herd immunity? We do, because we can’t stay locked down indefinitely, and presumably our rate will go up when we relax restrictions.
- jaynetics 6y ago
  R0 is the reproduction rate without any countermeasures.
  - lostlogin 6y ago
    Thanks, this is the key.
- Izkata 6y ago
  If reproduction rate stays under 1, it will eventually completely die out on its own.
- SpicyLemonZest 6y ago
  In this kind of modeling, outbreaks are considered to be impossible once transmission is driven below 1. (The herd immunity level is just the level at which the transmission rate is driven below 1 without additional interventions.)
dr_dshiv 6y ago
So, when active spreaders do most of the spreading, when they have herd immunity, there is a disproportionate affect on the overall spread. Meaning, let the young and stupid go out and play, it will allow for herd immunity at a lower rate than the typical randomly distributed rate.
throwaway122378 6y ago
Can anyone explain this in layman’s terms
- Ciantic 6y ago
  The paper says that herd immunity can be achived easier when the disease itself does the spreading. Because in "classical" herd immunity the vaccination does not target the active population so well. Instead when the active population gets the virus naturally it will produce stronger herd immunity.
  - e40 6y ago
    Basically swedish model may be the right one. What is that again?
    - Ciantic 6y ago
      Okay I removed that part, but Tegnell has now and then suggested the Swedish model is "herd immunity" and then sometimes suggested it's not. It does indicate the Swedish model could work, because the paper says it's just 43% with "disease-induced" herd immunity. And according to Swedish authorities 25% of Stockholm population already has antibodies.
      - Thlom 6y ago
        As I understand the Swedes, Herd immunity is not the goal they are working towards, but a possible bi-product of their strategy. The strategy as such is to keep the curve low enough, pretty much the same as everywhere else. They've just used less extreme measures to achieve this because they don't think it will matter in the long run.
        stareatgoats 6y ago
        That is, if you want to give them credit for any strategy at all. And no, the curve is not the same as everywhere [0]. Being a resident Swede, my lasting impression is that the so-called "strategy" has two legs (can be backed by a long list of official statements) 1: hope that the virus does not hit, and if it does, that it is not necessary to do anything and/or it is already too late anyway and that hopefully it will disappear through herd immunity. A hope that has been expressed up to twice weekly for the last 2 months or so, while the number of deaths go up and up. And 2: in a manner typical of governmental agencies everywhere: never ever admit to any mistake or error in judgement. Like admitting that the recommendation for the public not to use face masks was wrong, and was brought on by secondary concerns (over availability in hospitals). Even now (as of yesterday) the official stance is that using face masks might increase risks and efficacy is not backed by "scientific proof". The paper should probably be seen in this light. The main author of the paper Tom Britton forms together with Giesecke and Tegnell the "herd immunity" triumvirate that is responsible for the Swedish mess that currently renders Sweden a place among the top 10 countries with most fatalities per million. At the face of it this paper just seems to confirm that protective measures actually work. But by shining the light the herd immunity aspect of the equation he makes it sound like they were right all along. [0] https://mackuba.eu/corona/#compare?val=d&align100=1&pop=1&c=se,fi,no,tw,us https://mackuba.eu/corona/#compare?val=d&align100=1&pop=1&c=...
claudeganon 6y ago
All this talk of herd immunity is not terribly helpful because we don’t know what “immunity” means in practical terms. The duration of immunity for other corinaviruses varies quite a bit, the disease has the potential to cause permanent respiratory damage, with uncertain consequences for the possibility of reinfection, and we’re still seeing novel symptoms arise as in the case of the children with Kawasaki-like inflammation in NYC. I get the strong desire to return to a normal state of affairs, but paving over reality with scientific notions, deployed pseudoscientifically, does very little to get us there.
- SpicyLemonZest 6y ago
  We know that the vast majority of people will recover fully, and after they recover they'll be immune for a long time. Media reports have severely overstated the uncertainty here; if coronavirus can indeed trigger Kawasaki syndrome, it's only in the same sense that acetaminophen can cause liver damage.
  - macintux 6y ago
    > We know that the vast majority of people will recover fully, and after they recover they'll be immune for a long time. How do we “know” either of those facts? You may well be right, but I don’t see a scientific consensus yet.
    - SpicyLemonZest 6y ago
      A combination of animal studies, observation of recovered patients, and some studies directly on the antibodies recovered patients produce. In the US at least, public health authorities have consistently said from the start that there's substantial long-lasting immunity.
      - claudeganon 6y ago
        This is not in anyway true and I’m not sure why you’re suggesting it is. Dr. Deborah Birx has said expressly that we don’t know the duration of immunity and initial reviews suggest immunity might not last as long as we’d hoped: https://www.technologyreview.com/2020/04/27/1000569/how-long-are-people-immune-to-covid-19/ https://www.technologyreview.com/2020/04/27/1000569/how-long...
mrfusion 6y ago
My question is what R0 looks like in a city with mass transit vs without. The subway in NYC is huge and puts so many people together in close contact every day. Is it outrageous to think R0 could be 3.5 in NYC but 1.5 somewhere else?
- mannykannot 6y ago
  I think that, in the terminology of the paper, a higher proportion of New Yorkers would be considered 'high-active' compared to, say, LA, and perhaps there is a level that you don't even see in less crowded-mobile societies. I would guess it is helpful to keep the biological and social factors separate when modeling what is going on.
- gns24 6y ago
  > Is it outrageous to think R0 could be 3.5 in NYC but 1.5 somewhere else? No, not at all. It would be outrageous to think that it doesn't vary by a factor of two IMO.
code_duck 6y ago
"lower than the classical level" would be a nice way to make the submission title make sense and be a complete sentence.
mrfusion 6y ago
This is great news!
- permalac 6y ago
  Without peer review this is weak.
octonion 6y ago
This is also true under much weaker assumptions. Here's a simple example calculation. Say 50% of a population has R0=1.5, 50% has R0=2.5; the average R0 is 2. Then the required herd immunity is (2-1/1.5-1/2.5)/2=0.47, below the classical 1-1/2=0.5. That this always true for any number of independent subpopulations is a consequence of the harmonic-arithmetic mean inequality.

34 points by 2a0c40 6y ago

Interesting that it seems to suggest there is an optimal amount of distancing/lockdown/etc. to lower the peak, but avoid a big "rebound". I don't see a lot of discussion about how to find that optimum amount. It has occurred to me that the planet is essentially conducting a gigantic experiment in virus evolution, by creating a novel viral selection environment never seen before. For example, how does it impact a virus if the strains most likely to send the host to the hospital, actually spread more than the strains most likely to be asymptomatic, because even asymptomatic hosts are in isolation? Unless there is a vaccine or cure coming soon (there isn't), then the end result must be herd immunity, no matter what the policy. It would seem that we should be considering how to select for the healthiest segment of the population to provide that herd immunity, rather than trying to isolate as much as possible in the vain hope that it will die out prior to that point. If that was every possible, it has long since spread way too far to expect that to be possible. There might be cases (e.g. MERS) where you can stomp out a virus before it spreads enough to cause herd immunity, but we have long since passed that point. I don't think most people (including policy makers) are thinking about the endgame properly (as this paper does).

jakeogh 6y ago
Senegal is worth looking at (end of vid): https://youtu.be/iBma_0oAiMI https://youtu.be/iBma_0oAiMI
macintux 6y ago
As far as I can tell, there is one big unknown that makes planning particularly difficult: What are the long-term health effects from being infected? If it turns out that some significant percentage of the population is going to have a negative impact over the course of their lives from this, should we continue to hunker down until a vaccine or better treatments are found?
- irq11 6y ago
  This is not an unknown. We have very good data on outcomes. The media has been diving ever-further into niche stories of bad outcomes in order to drive clicks, but the numbers of cases driving these narratives are vanishingly small. For the vast majority (>98%) of those who are under 60 and otherwise healthy, this disease is on par with the flu. Even amongst those who are more vulnerable, the rate of long-term sequelae is low. It’s the people who end up in the ICU who are experiencing the most long-term effects, and those people are a single-digit percentage of a single-digit percentage of those infected.
  - macintux 6y ago
    This disease has been here for 6 months. The degree of certainty in your assessment seems unwarranted. I want to believe it, but optimism and certainty given the novelty of this virus seem strange bedfellows.
    - xamuel 6y ago
      As rossdavidh correctly pointed out, if we don't get a vaccine soon (which we probably won't), then herd immunity is virtually the only way out anyway. Even if it turns out there are long term health problems, that doesn't really change that calculus. Very few people can completely isolate themselves indefinitely (remember, the virus doesn't just magically not spread in supermarkets), and if everyone did that, most of us would starve and the few who didn't starve would inherit a post-civilization ruin.
      - mlyle 6y ago
        It does change the economic break even point a little bit. If intervention A reduces infection count 10% in the time to a vaccine, whether it's worth it depends upon whether it prevents 20,000 deaths, or 20,000 deaths and 40,000 long term illnesses. The threshold to herd immunity depends upon contact rate. So interventions well short of "lockdown" may not completely prevent the spread of the virus but still reach equilibrium at a lower number of people infected.
Isinlor 6y ago
Currently we need to learn more about this virus to make informed decisions - there is more we still don't know, than we do know for sure. Trying to slowly contain the virus like in South Korea, Taiwan, New Zealand is a strategy worth a shot for most of the western world, unless we learn something dramatic new about it. There is no need for draconian lock downs after exponential spread is stopped, but we do need good track and trace programs, masks etc. There are very high risks associated with Swedish strategy of either killing a lot of people due to bad separation of vulnerable people, destroying health of the big part of the population (see below), or even just failing to reach herd immunity because social distancing, closure of schools, and ban on mass gathering will be good enough to slow down the spread a lot, but bad enough to kill disproportionate amount of people. For example Sweden has many times more deaths than other Nordic Countries, but similar economical outlooks, and the even the idea of reaching herd immunity is uncertain. It may be that other Nordic countries will have soon less or about the same amount of restrictions as Sweden, but many times less deaths. And regarding the risks, one of the scientists who discovered Ebola [0]: "Many people think COVID-19 kills 1% of patients, and the rest get away with some flulike symptoms. But the story gets more complicated. Many people will be left with chronic kidney and heart problems. Even their neural system is disrupted. There will be hundreds of thousands of people worldwide, possibly more, who will need treatments such as renal dialysis for the rest of their lives. The more we learn about the coronavirus, the more questions arise. We are learning while we are sailing. That’s why I get so annoyed by the many commentators on the sidelines who, without much insight, criticize the scientists and policymakers trying hard to get the epidemic under control. That’s very unfair." Also based on Japanese study on around 100 COVID-19 cases from Grand Princess passengers [1] and some 6 divers all with mild cases from Germany [2] around 50 to 80% of people seem to have damage to the lungs visible on CT scans after mild symptomatic or even asymptomatic infection. Two of the divers had significant oxygen deficiency when doing physical exercise after 5-6 weeks of recovery. The guess of the doctor who was inspecting them is that this is permanent damage that may take years to recover if ever. [0] https://www.sciencemag.org/news/2020/05/finally-virus-got-me-scientist-who-fought-ebola-and-hiv-reflects-facing-death-covid-19 https://www.sciencemag.org/news/2020/05/finally-virus-got-me... [1] https://pubs.rsna.org/doi/10.1148/ryct.2020200110 https://pubs.rsna.org/doi/10.1148/ryct.2020200110 [2] https://translate.google.com/translate?sl=auto&tl=en&u=https%3A%2F%2Fwww.wetnotes.eu%2Ftauchen-nach-covid-19-erkrankung%2F https://translate.google.com/translate?sl=auto&tl=en&u=https...
deleted 6y ago
[deleted]

medymed 6y ago

It seems like their simulated model makes certain members of a population less ‘active’ and so at less parametric risk of infection/transmission than in a model where everyone is active. This is useful, and/but doesn’t appear to be Covid-19 specific.

SpicyLemonZest 6y ago
It isn't, and the same phenomenon has been observed in other cases. The 2009 swine flu for example had R0 estimated between 1.4-1.6, yielding a theoretical herd immunity threshold of 29%-33%, but estimates of how many people were actually infected range from 11%-21%.
jaynetics 6y ago
I thought their point was that highly active people will have been infected in greater numbers after a lockdown, numbers that are even more divergent than without a lockdown, and thus less other people will need to get infected.

lostlogin 6y ago

> The classical herd immunity level hC is defined as hC=1−1/R0 How do you use this in places where transmission is less than 1? New Zealand now has a transmission rate considerably under 1, but this seems to break the calculation. Does this mean that a reproduction number under 1 doesn’t need herd immunity? We do, because we can’t stay locked down indefinitely, and presumably our rate will go up when we relax restrictions.

jaynetics 6y ago
R0 is the reproduction rate without any countermeasures.
- lostlogin 6y ago
  Thanks, this is the key.
Izkata 6y ago
If reproduction rate stays under 1, it will eventually completely die out on its own.
SpicyLemonZest 6y ago
In this kind of modeling, outbreaks are considered to be impossible once transmission is driven below 1. (The herd immunity level is just the level at which the transmission rate is driven below 1 without additional interventions.)

dr_dshiv 6y ago

So, when active spreaders do most of the spreading, when they have herd immunity, there is a disproportionate affect on the overall spread. Meaning, let the young and stupid go out and play, it will allow for herd immunity at a lower rate than the typical randomly distributed rate.

throwaway122378 6y ago

Can anyone explain this in layman’s terms

Ciantic 6y ago
The paper says that herd immunity can be achived easier when the disease itself does the spreading. Because in "classical" herd immunity the vaccination does not target the active population so well. Instead when the active population gets the virus naturally it will produce stronger herd immunity.
- e40 6y ago
  Basically swedish model may be the right one. What is that again?
  - Ciantic 6y ago
    Okay I removed that part, but Tegnell has now and then suggested the Swedish model is "herd immunity" and then sometimes suggested it's not. It does indicate the Swedish model could work, because the paper says it's just 43% with "disease-induced" herd immunity. And according to Swedish authorities 25% of Stockholm population already has antibodies.
    - Thlom 6y ago
      As I understand the Swedes, Herd immunity is not the goal they are working towards, but a possible bi-product of their strategy. The strategy as such is to keep the curve low enough, pretty much the same as everywhere else. They've just used less extreme measures to achieve this because they don't think it will matter in the long run.
      - stareatgoats 6y ago
        That is, if you want to give them credit for any strategy at all. And no, the curve is not the same as everywhere [0]. Being a resident Swede, my lasting impression is that the so-called "strategy" has two legs (can be backed by a long list of official statements) 1: hope that the virus does not hit, and if it does, that it is not necessary to do anything and/or it is already too late anyway and that hopefully it will disappear through herd immunity. A hope that has been expressed up to twice weekly for the last 2 months or so, while the number of deaths go up and up. And 2: in a manner typical of governmental agencies everywhere: never ever admit to any mistake or error in judgement. Like admitting that the recommendation for the public not to use face masks was wrong, and was brought on by secondary concerns (over availability in hospitals). Even now (as of yesterday) the official stance is that using face masks might increase risks and efficacy is not backed by "scientific proof". The paper should probably be seen in this light. The main author of the paper Tom Britton forms together with Giesecke and Tegnell the "herd immunity" triumvirate that is responsible for the Swedish mess that currently renders Sweden a place among the top 10 countries with most fatalities per million. At the face of it this paper just seems to confirm that protective measures actually work. But by shining the light the herd immunity aspect of the equation he makes it sound like they were right all along. [0] https://mackuba.eu/corona/#compare?val=d&align100=1&pop=1&c=se,fi,no,tw,us https://mackuba.eu/corona/#compare?val=d&align100=1&pop=1&c=...

claudeganon 6y ago

All this talk of herd immunity is not terribly helpful because we don’t know what “immunity” means in practical terms. The duration of immunity for other corinaviruses varies quite a bit, the disease has the potential to cause permanent respiratory damage, with uncertain consequences for the possibility of reinfection, and we’re still seeing novel symptoms arise as in the case of the children with Kawasaki-like inflammation in NYC. I get the strong desire to return to a normal state of affairs, but paving over reality with scientific notions, deployed pseudoscientifically, does very little to get us there.

SpicyLemonZest 6y ago
We know that the vast majority of people will recover fully, and after they recover they'll be immune for a long time. Media reports have severely overstated the uncertainty here; if coronavirus can indeed trigger Kawasaki syndrome, it's only in the same sense that acetaminophen can cause liver damage.
- macintux 6y ago
  > We know that the vast majority of people will recover fully, and after they recover they'll be immune for a long time. How do we “know” either of those facts? You may well be right, but I don’t see a scientific consensus yet.
  - SpicyLemonZest 6y ago
    A combination of animal studies, observation of recovered patients, and some studies directly on the antibodies recovered patients produce. In the US at least, public health authorities have consistently said from the start that there's substantial long-lasting immunity.
    - claudeganon 6y ago
      This is not in anyway true and I’m not sure why you’re suggesting it is. Dr. Deborah Birx has said expressly that we don’t know the duration of immunity and initial reviews suggest immunity might not last as long as we’d hoped: https://www.technologyreview.com/2020/04/27/1000569/how-long-are-people-immune-to-covid-19/ https://www.technologyreview.com/2020/04/27/1000569/how-long...

mrfusion 6y ago

My question is what R0 looks like in a city with mass transit vs without. The subway in NYC is huge and puts so many people together in close contact every day. Is it outrageous to think R0 could be 3.5 in NYC but 1.5 somewhere else?

mannykannot 6y ago
I think that, in the terminology of the paper, a higher proportion of New Yorkers would be considered 'high-active' compared to, say, LA, and perhaps there is a level that you don't even see in less crowded-mobile societies. I would guess it is helpful to keep the biological and social factors separate when modeling what is going on.
gns24 6y ago
> Is it outrageous to think R0 could be 3.5 in NYC but 1.5 somewhere else? No, not at all. It would be outrageous to think that it doesn't vary by a factor of two IMO.

code_duck 6y ago

"lower than the classical level" would be a nice way to make the submission title make sense and be a complete sentence.

mrfusion 6y ago

This is great news!

permalac 6y ago
Without peer review this is weak.

octonion 6y ago

This is also true under much weaker assumptions. Here's a simple example calculation. Say 50% of a population has R0=1.5, 50% has R0=2.5; the average R0 is 2. Then the required herd immunity is (2-1/1.5-1/2.5)/2=0.47, below the classical 1-1/2=0.5. That this always true for any number of independent subpopulations is a consequence of the harmonic-arithmetic mean inequality.